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On the wikipedia page in my native language it is stated that according to the central limit theorem for $X_i$ iid with finite mean and variance we have that $\bar{X}_n$ is approximately normal. Isn’t this wrong? Take for example $X_i$ bernoulies then $0<\bar{X}_n<1$. So it can’t be normally distributed right?

I always thought that it only holds for the rescaled average. Can someone help me out here?

Dumbboi
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  • The classical central limit theorem says something about the distribution of $Z_n=\frac{\overline{X}_n-\mu}{\sigma/\sqrt{n}}$ in the limit as $n\to\infty$. However it says nothing explicit about how good that approximation may be at any finite sample size. Indeed, 'how close is close enough' very much depends on what you're doing; for many purposes it may be fine in the middle of the distribution but not great in the tails, so if you're relying on the tails you may need a much larger $n$. ... ctd – Glen_b Sep 06 '22 at 04:57
  • ctd... The closeness you get (in cdf terms) also depends on the distribution you started with. In some cases it's excellent at very small sample size (like n=4 or 5 say), and in others it's poor even at huge sample sizes (like n in the millions). Note that if you don't know the parameters, there are other issues involved besides the CLT. There are many posts on site that discuss the central limit theorem, so a few judicious searches may be helpful – Glen_b Sep 06 '22 at 04:58
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    The Bernoulli is constrained to {0,1} but $Z_n$ is not, and in the limit its support will pass any finite bound. Or looked at another way, as $n$ increases, $0$ and $1$ will be more and more standard errors away from the mean, so the bounds will be of decreasing concern as $n$ gets large. – Glen_b Sep 06 '22 at 05:04

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